Antimagic and magic labelings in Cayley digraphs - Semantic Scholar

AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 55 (2013), Pages 65–71

Antimagic and magic labelings in Cayley digraphs T. Tamizh Chelvam

N. Mohamed Rilwan

G. Kalaimurugan Department of Mathematics Manonmaniam Sundaranar University Tirunelveli 627 012, Tamil Nadu India [email protected] [email protected] [email protected] Abstract A Cayley digraph is a digraph constructed from a group Γ and a generating subset S of Γ. It is denoted by CayD (Γ, S). In this paper, we prove for any finite group Γ and a generating subset S of Γ, that CayD (Γ, S) admits a super vertex (a, d)-antimagic labeling depending on d and |S|. We provide algorithms for constructing the labelings.

1

Introduction

A graph labeling is an assignment of integers to the vertices or edges or both, subject to certain conditions. MacDougall et al. [3] introduced the notion of a vertex-magic total labeling. A digraph G = (V, E) with |V | = p and |E| = q is called a (p, q) digraph. For a graph G = (V, E), a one-to-one mapping f : V ∪ E → {1, 2, . . . , |V | + |E|} is a vertex-magic total labeling if there is a constant k, called the magic constant, such that for every vertex v, f (v)+ uv∈E f (uv) = k where the sum is over all vertices u adjacent to v. Thirusangu et al. [4] studied the super vertex (a, d)-antimagic and vertex-magic total labelings for a particular class of Cayley digraphs. For a survey of magic and antimagic labelings one can refer to Gallian [1]. A (p, q)-digraph G = (V, E) is said to have a super vertex (a, d)-antimagic labeling if there exists a function f : V ∪ E → {1, 2, . . . , p + q} such that f (V ) = {1, 2, . . . , p} and the sum of the labels of the outgoing arcs of v and its own label are distinct for all v ∈ V . Moreover, the set of all such distinct labels associated with vertices in V of G is {a, a + d, a + 2d, . . . , a + (p − 1)d}, where a and d are two positive integers. A (p, q)-digraph G is said to be (0, 1)-vertex-magic with common vertex count k if there exists a bijection f : E(G) → {1, 2, . . . , q} such that for each u ∈ V (G),

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T.T. CHELVAM, N.M. RILWAN AND G. KALAIMURUGAN

f (e) = k where E(u) is the set of all outgoing arcs from u. It is said to be (0, 1)-vertex-antimagic if, for each u ∈ V (G), the sums f (e) are distinct.

e∈E(u)

e∈E(u)

Let Γ be a finite nontrivial group and S be a generating subset of Γ. The Cayley digraph G, CayD (Γ, S), is the digraph whose vertices are the elements of Γ, and there is an arc from α to ασ whenever α ∈ Γ and σ ∈ S. If S = S −1 , then there is an arc from α to ασ if and only if there is an arc from ασ to α. Cayley digraphs are useful designs for many well-known interconnection networks. For example, rings, tori, hypercubes, butterflies, cubic-connected cycle networks, are Cayley digraphs [2]. In this paper, we prove that G = CayD (Γ, S) admits a super vertex (a, d)antimagic labeling for d = 1 when |S| is even, and for d = 2 when |S| is odd. Also CayD (Γ, S) admits a (0, 1)-vertex-magic labeling when |S| is even, whereas the same admits a (0, 1)-vertex-antimagic labeling and a vertex-magic total labeling when |S| is odd.

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Super vertex (a, d)-antimagic labeling of Cayley digraphs

In this section, we give an algorithm to construct a super vertex (a, d)-antimagic labeling of the Cayley digraph CayD (Γ, S). The algorithm produces a super vertex (a, 1)-antimagic labeling if |S| is even and a super vertex (a, 2)-antimagic labeling if |S| is odd. Algorithm 1: Super vertex (a, d)-antimagic labeling of CayD (Γ, S) Input: A finite group Γ = {v1 , v2 , . . . , vm } and a generating subset S = {σ1 , σ2 , . . . , σ|S| }. Denote the vertices and arcs of the Cayley digraph G = CayD (Γ, S) as V (G) = {v1 , v2 , . . . , vm} and E(G) = {eij : eij = vi .σj , 1 ≤ i ≤ m, 1 ≤ j ≤ |S|}. Procedure: Define f : V (G) ∪ E(G) −→ {1, 2, . . . , |V (G)| + |E(G)|} by f (vi ) = i, for 1 ≤ i ≤ m, and (j + 1)m + 1 − i if j is even, f (eij ) = jm + i if j is odd, for 1 ≤ i ≤ m and 1 ≤ j ≤ |S|. Output: The sum corresponding to the vertex vi is given by f (vi ) +

|S| j=1

f (eij ) =

|S| 2 [(|S|+ 2)m + 1] + i |S|−1 [(|S| + 1)m + 1] + |S|m + 2i 2

if |S| is even, if |S| is odd.

Example 2.1 We illustrate Algorithm 1 for CayD (Z6 , S), where S = {1, 2, 4, 5} ⊂ Z6 . Actually we give a super vertex (75, 1)-antimagic labeling of CayD (Z6 , {1, 2, 4, 5}) in Figure 1.

LABELINGS IN CAYLEY DIGRAPHS

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J HH J 28 12 13 21 HH

] Y * HJ H r Jr3 6

H 16 24 JH

j H ^J HH

25J 9

H HH J

HH J

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HH 6 23 6 27 15 H * Y H H r

11J]J r HH

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26

10 4

Figure 1 Theorem 2.2 Let Γ be a finite group of order m and let S be a generating subset of Γ. Then the Cayley digraph CayD (Γ, S) admits a super vertex (a, 1)-antimagic labeling if |S| is even and a super vertex (a, 2)-antimagic labeling if |S| is odd. Proof: Let Γ = {v1 , v2 , . . . , vm } and S = {σ1 , σ2 , . . . , σ|S| }. By the construction of G = CayD (Γ, S), V (G) = {v1 , v2 , . . . , vm } and E(G) = {eij : eij = vi .σj , 1 ≤ i ≤ m, 1 ≤ j ≤ |S|}. Let us consider the assignment of labels as per Algorithm 1 and |S| realize that f (vi ) + f (eij ) are distinct for every i, 1 ≤ i ≤ m. Note that the j=1

following are true: Case 1: When |S| is even, the set of numbers corresponding to various sums are [(|S| + 2)m + 1] + 1. Hence CayD (Γ, S) {a, a + 1, a + 2, . . . , a + (m − 1)} where a = |S| 2 admits a super vertex (a, 1)-antimagic labeling. Case 2: When |S| is odd, the set of numbers corresponding to various sums are |S|−1 {a, a + 2, a + 4, . . . , a + 2(m − 1)} where a = [(|S| + 1)m + 1] + |S|m + 2. 2 Hence CayD (Γ, S) admits a super vertex (a, 2)-antimagic labeling. By taking Γ as the Dihedral group Dn = r, s | rn = s2 = e, rsr = s and S = {r, s}, we get the following: Corollary 2.3 [4, Theorem 3.1] The Cayley digraph associated with the group Dn , with the generating set {r, s}, admits a super vertex (a, d)-antimagic labeling.

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Vertex-magic total labeling of Cayley digraphs

In this section, we give an algorithm to construct a vertex-magic total labeling of the Cayley digraph CayD (Γ, S) when |S| is odd. Also we give an algorithm to construct a (0, 1)-vertex-magic and a (0, 1)-vertex-antimagic labeling of CayD (Γ, S).

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Algorithm 2: Vertex-magic total labeling of CayD (Γ, S) Input: A finite group Γ = {v1 , v2 , . . . , vm } and a generating subset S = {σ1 , σ2 , . . . , σ|S| }. Denote the vertices and arcs of the Cayley digraph G = CayD (Γ, S), by V (G) = {v1 , v2 , . . . , vm} and E(G) = {eij : eij = vi .σj , 1 ≤ i ≤ m, 1 ≤ j ≤ |S|}. Procedure: Define f : V (G) ∪ E(G) −→ {1, 2, . . . , |V (G)| + |E(G)|} by f (vi ) = i, for each i, 1 ≤ i ≤ m. Further, jm + i if j is even, f (eij ) = (j + 1)m + 1 − i if j is odd, for 1 ≤ i ≤ m and 1 ≤ j ≤ |S|. Output: The sum corresponding to the vertex vi is given by |S| |S| [(|S| + 2)m + 1] + i if |S| is even, 2 f (eij ) = m(|S|+1) f (vi ) + 2 + |S|+1 if |S| is odd. 2 2 j=1 Theorem 3.1 Let Γ be a finite group of order m and S be a generating subset of Γ such that |S| is odd. Then the Cayley digraph CayD (Γ, S) admits a vertex-magic total labeling. Proof: Let us take Γ = {v1 , v2 , . . . , vm } and S = {σ1 , σ2 , . . . , σ|S| }, where |S| is odd. By the construction of G = CayD (Γ, S), E(G) = {eij : eij = vi .σj , 1 ≤ i ≤ m, 1 ≤ j ≤ |S|}. Consider the assignment of labels by Algorithm 2. Since |S| is odd, for |S| 2 each vertex vi , the sum f (vi ) + f (eij ) = m(|S|+1) + |S|+1 is the same. 2 2 j=1

Remark 3.2 If |S| is even, then Algorithm 2 produces a super vertex (a, 1)antimagic labeling of CayD (Γ, S). Example 3.3 We illustrate Algorithm 2 for CayD (D4 , S), where D4 = r, s | r4 = s2 = e, rsr = s and S = {r, r3 , s}. Actually we exhibit a vertex-magic total labeling of CayD (D4 , {r, r3 , s}) with total sum 66 in Figure 2. Let Γ = Zn = {0, 1, . . . , n−1} and S = {a, b, b+k} ⊆ Zn such that gcd(a, b, k) = 1 and any one of the following holds: (i) gcd(a − b, k) = 1, (ii) gcd(a, 2k) = 1, (iii) gcd(b, k) = 1,

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(iv) both a and k are even, (v) a is odd and either b or k is odd.

6 22 285r 12 r 9 9 27 21 ? 11 ? 32 29 1 4 : : r r 17

13

16 ?

20 ? 23

9 6 6 10 25 r r 9 9 8 24 7 26 18 6 14 6 31 30 : : r r 2

15

19

3 Figure 2

Note that |S| is odd and S is a generating set for Zn . By Theorem 3.1, we have the following: Corollary 3.4 [4, Theorem 4.1] The Cayley digraph associated with the group Zn , with the generating set {a, b, b + k}, admits a magic labeling. Now let us present an algorithm for a (0, 1)-vertex-magic labeling and a (0, 1)vertex-antimagic labeling of the Cayley digraph CayD (Γ, S). Algorithm 3: (0, 1)-vertex-magic labeling and (0, 1)-vertex-antimagic labeling of CayD (Γ, S) Input: A finite group Γ = {v1 , v2 , . . . , vm } and a generating subset S = {σ1 , σ2 , . . . , σ|S| }. Denote the vertices and arcs of the Cayley digraph G = CayD (Γ, S) by V (G) = {v1 , v2 , . . . , vm } and E(G) = {eij : eij = vi .σj , 1 ≤ i ≤ m, 1 ≤ j ≤ |S|}. Procedure: Define f : E(G) −→ {1, 2, . . . , |E(G)|} by jm + 1 − i if j is even, f (eij ) = (j − 1)m + i if j is odd, for i, j with 1 ≤ i ≤ m and 1 ≤ j ≤ |S|. Output: The sum corresponding to the vertex vi is given by |S| m|S|2 + |S| if |S| is even, 2 2 f (eij ) = m(|S−1|) 2 |S|−1 + + (|S| − 1)m + i if |S| is odd. 2 2 j=1

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Example 3.5 We illustrate Algorithm 3 for CayD (D6 , {r, r2 , s}). Actually we exhibit a (0, 1)-vertex-antimagic labeling of CayD (D6 , {r, r2 , s}) in Figure 3. s 36 r@ 9 B 7 B @ 12 18 25 B @@ e : r @ B I 1 sr B@ R @ r @rsr5 B 13 31 9 9 B@ B B 35 @ B24 B B 8 6 N B ? @ B B17 B 30 26 : : 5 B @ r BNB B r r - B r B 19 B B B ? B B B 2 B B B14 B

B 23 B BM 11 B B B B6 B 2 B 16 sr B 4 r B 20 BB rsr 9 9 B @ B R @ 32 9 @B 34 B 5 6 B B 15 @ B 29B : : B 4 r B r r @B 22 r2 27 @ B @Br10 3 9 @ 21BBM 3 33 sr @ B @ B : I @Br 4@ r3

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Figure 3

In view of Algorithm 3, we have the following theorem.

Theorem 3.6 Let S be a generating subset of a finite group Γ and let G = CayD (Γ, S). Then the following hold: (i) If |S| is even, G admits a (0, 1)-vertex-magic labeling; (ii) If |S| is odd, G admits a (0, 1)-vertex-antimagic labeling.

Acknowledgments The authors express their sincere thanks to the referees for vital comments which improved the presentation of the paper. The work reported here is supported by the Special Assistance Programme (F510-DRS-I/2007) of University Grants Commission (UGC), India, awarded to the Department of Mathematics, Manonmaniam Sundaranar University for the period 2007–2012 to the first author. Also the work reported here is supported by the Basic Science Research Fellowship (BSRF–UGC) to the second and third authors.


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References [1] J.A. Gallian, A dynamic survey of graph labeling, Electron. J. Combin. 18 (2011), #DS6. [2] M. Heydemann, Cayley graphs and interconnection networks, In: G. Hahn and G. Sabidussi (Eds.), Graph Symmetry: Algebraic Methods and Applications (1997), 167–224. [3] J.A. MacDougall, M. Miller, Slamin and W.D. Wallis, Vertex-magic total labelings of graphs, Util. Math. 61 (2002), 3–21. [4] K. Thirusangu , A.K. Nagar and R. Rajeswari, Labelings in Cayley digraphs, European J. Combin. 32 (1) (2011), 133–139.

(Received 23 Jan 2011; revised 3 Aug 2012)